It’s like some people made stuff up just to confuse those paying attention to reality.
The origins are likely found in a letter by Bellarmine noted here which says things that are quite close:
"First. I say that it seems to me that Your Reverence and Galileo did prudently to content yourself with speaking hypothetically, and not absolutely, as I have always believed that Copernicus spoke. For to say that, assuming the earth moves and the sun stands still, all the appearances are saved better than with eccentrics and epicycles, is to speak well; there is no danger in this, and it is sufficient for mathematicians. But to want to affirm that the sun really is fixed in the center of the heavens and only revolves around itself (i. e., turns upon its axis ) without traveling from east to west, and that the earth is situated in the third sphere and revolves with great speed around the sun, is a very dangerous thing, not only by irritating all the philosophers and scholastic theologians, but also by injuring our holy faith and rendering the Holy Scriptures false. For Your Reverence has demonstrated many ways of explaining Holy Scripture, but you have not applied them in particular, and without a doubt you would have found it most difficult if you had attempted to explain all the passages which you yourself have cited."
Note that critical parts here are left out when the quote is used. Bellarmine appeals to scientific authority (mathematicians) and also appeals to the need to exegete passages properly (more in section 3).
"Second. I say that, as you know, the Council [of Trent] prohibits expounding the Scriptures contrary to the common agreement of the holy Fathers. And if Your Reverence would read not only the Fathers but also the commentaries of modern writers on Genesis, Psalms, Ecclesiastes and Josue, you would find that all agree in explaining literally (ad litteram) that the sun is in the heavens and moves swiftly around the earth, and that the earth is far from the heavens and stands immobile in the center of the universe. Now consider whether in all prudence the Church could encourage giving to Scripture a sense contrary to the holy Fathers and all the Latin and Greek commentators. Nor may it be answered that this is not a matter of faith, for if it is not a matter of faith from the point of view of the subject matter, it is on the part of the ones who have spoken. It would be just as heretical to deny that Abraham had two sons and Jacob twelve, as it would be to deny the virgin birth of Christ, for both are declared by the Holy Ghost through the mouths of the prophets and apostles."
"Third. I say that if there were a true demonstration that the sun was in the center of the universe and the earth in the third sphere, and that the sun did not travel around the earth but the earth circled the sun, then it would be necessary to proceed with great caution in explaining the passages of Scripture which seemed contrary, and we would rather have to say that we did not understand them than to say that something was false which has been demonstrated. But I do not believe that there is any such demonstration; none has been shown to me. It is not the same thing to show that the appearances are saved by assuming that the sun really is in the center and the earth in the heavens. I believe that the first demonstration might exist, but I have grave doubts about the second, and in a case of doubt, one may not depart from the Scriptures as explained by the holy Fathers. I add that the words ' the sun also riseth and the sun goeth down, and hasteneth to the place where he ariseth, etc.' were those of Solomon, who not only spoke by divine inspiration but was a man wise above all others and most learned in human sciences and in the knowledge of all created things, and his wisdom was from God. Thus it is not too likely that he would affirm something which was contrary to a truth either already demonstrated, or likely to be demonstrated. And if you tell me that Solomon spoke only according to the appearances, and that it seems to us that the sun goes around when actually it is the earth which moves, as it seems to one on a ship that the beach moves away from the ship, I shall answer that one who departs from the beach, though it looks to him as though the beach moves away, he knows that he is in error and corrects it, seeing clearly that the ship moves and not the beach. But with regard to the sun and the earth, no wise man is needed to correct the error, since he clearly experiences that the earth stands still and that his eye is not deceived when it judges that the moon and stars move. And that is enough for the present. I salute Your Reverence and ask God to grant you every happiness."
The third point is the most critical, because it reveals that contrary to how the quotes are used, Bellarmine was ready to submit to the possibility that heliocentrism was true and interpret the text accordingly, and that he also appealed to scientific authority (of Solomon, however misplaced we may think this is) as well as to observation. In his view, this was a case of "extraordinary claims require extraordinary evidence" -- and surely Skeptics can respect that (especially since so many astronomers of the day accepted geocentrism).
Source: http://www.tektonics.org/af/bogusq.php
Bellarmine added that if heliocentrism could be conclusively, scientifically proven: “then one would have to proceed with great care in explaining the Scriptures that appear contrary; and say rather that we do not understand them, than that what is demonstrated is false.”
Too many people think the Galileo case proves that the Church should bow to science without hesitation. Not at all. It proves exactly the opposite—that the Church should NOT bow to science uncritically, because that’s exactly how the Church got itself into the whole mess.
The Ptolemaic model came from Claudius Ptolemy, who was a scientist, not a churchman. His was an accepted scientific theory for more than a thousand years and worked quite well in predicting the motions of the planets. The Church eventually—it didn’t really care about such matters in the beginning—took this well-established science and increasingly theologized it to the point that Dante could describe it in the Paradiso and Bellarmine could feel it impertinent to deny it.